May 15, 2010: This paper has been corrected and is being split into
two smaller papers. The first paper,
The Strengthened Hanna Neumann Conjecture I: A
Combinatorial Proof, is now available. In this paper the sheaf theory
has been written in purely combinatorial terms. A second paper is
in preparation, to describe the sheaf theory and give a very short proof
of the conjecture ("very short" assuming known cases of the conjecture and
assuming the sheaf theory is in place).
June 12, 2009: An error was found in the paper; we are trying to fix
the error and post an updated manuscript
before we distribute the paper further.

Postscript version.

Dvi version.

PDF version.

Abstract:

We prove the Strengthened Hanna Neumann Conjecture. We give a more direct
cohomological interpretation of the conjecture in terms of "typical" covering
maps, and use graph Galois theory to "symmetrize" the conjecture. The
conjecture is then related to certain kernel of a morphism of sheaves, and is
implied provided these kernels are co-acyclic in the covering cohomology
theory. This allows us to prove a slightly generalized Strengthened Hanna
Neumann Conjecture; this conjecture is false if generalized to all sheaves.
The kernels we use do not exist in the theory of graphs, so our use of sheaf
theory seems essential to this approach.